Simplify Boolean Expressions

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Simplifying Boolean Expressions

Boolean expressions are built from variables and logical operations such as AND, OR, and NOT. The goal is to rewrite an expression into an equivalent, simpler form while keeping the same truth value for every input.

1) Identify the structure

First, separate the expression into smaller parts. Notice which parts are grouped by parentheses and which operation applies first. This helps you see where simplification can happen.

2) Apply basic Boolean laws

Use standard equivalences such as:

Identity: A AND true = A, A OR false = A
Domination: A OR true = true, A AND false = false
Idempotent: A OR A = A, A AND A = A
Complement: A OR NOT A = true, A AND NOT A = false
Double negation: NOT(NOT A) = A

3) Look for patterns

After simplifying inside parentheses, check whether some terms can be removed or combined. Distribute only when it helps, and use De Morgan’s laws when a negation applies to a whole grouped expression.

4) Check your result

Test the simplified form on a few input cases, or compare it with the original expression using a truth table if needed. The two expressions should always match.

A good final answer is the shortest equivalent expression that preserves the original logic.